## Document Type

Article

## Publication Date

8-6-2024

## Abstract

In single-particle Madelung mechanics, the single-particle quantum state Ψ(⃗x, t) = R(⃗x, t)eiS(⃗x,t)/h is interpreted as comprising an entire conserved fluid of classical point particles, with local density R(⃗x, t)2 and local momentum ⃗∇S(⃗x, t) (where R and S are real). The Schrödinger equation gives rise to the continuity equation for the fluid, and the Hamilton–Jacobi equation for particles of the fluid, which includes an additional density-dependent quantum potential energy term Q(⃗x, t) = − ¯h2 2m ⃗∇R(⃗x,t) R(⃗x,t) , which is all that makes the fluid behavior nonclassical. In particular, the quantum potential can become negative and create a nonclassical boost in the kinetic energy. This boost is related to superoscillations in the wavefunction, where the local frequency of Ψ exceeds its global band limit. Berry showed that for states of definite energy E, the regions of superoscillation are exactly the regions where Q(⃗x, t) < 0. For energy superposition states with band-limit E+, the situation is slightly more complicated, and the bound is no longer Q(⃗x, t) < 0. However, the fluid model provides a definite local energy for each fluid particle which allows us to define a local band limit for superoscillation, and with this definition, all regions of superoscillation are again regions where Q(⃗x, t) < 0 for general superpositions. An alternative interpretation of these quantities involving a reduced quantum potential is reviewed and advanced, and a parallel discussion of superoscillation in this picture is given. Detailed examples are given which illustrate the role of the quantum potential and superoscillations in a range of scenarios.

## Recommended Citation

Mordecai Waegell 2024 *New J. Phys.* **26** 083002 https://doi.org/10.1088/1367-2630/ad689b

## Peer Reviewed

1

## Copyright

The author

## Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 License.

## Comments

This article was originally published in

New Journal of Physics, volume 26, in 2024. https://doi.org/10.1088/1367-2630/ad689b